Myths and Mysteries about mirrors.

There are many online web sites and many publications since long before the
world wide web existed, offering answers to the question:

Why do mirrors flip horizontally but not vertically?

An example is the answer sketched by Douglas Hofstadter in

The Mind's I: Fantasies and reflections on self and soul
(Edited by D. Hofstadter and D.C. Dennett)

Hofstadter wrote, in the "Reflections" section following Thomas Nagel's "What is
it like to be a bat?" (Chapter 24):

There is a famous puzzle in mathematics and physics courses. It asks,
"Why does a mirror reverse left and right, but not up and down?" It
gives many people pause for thought, and if you don't want to be told
the answer, skip the next two paragraphs. The answer hinges on what we
consider a suitable way to project ourselves onto our mirror images. Our
first reaction is that by walking forward a few steps and then spinning'
around on our heels, we could step into the shoes of "that person" there
in the mirror-forgetting that the heart, appendix, and so forth of "that
person" are on the wrong side. On a gross anatomical level, that image
is actually of a nonperson. Microscopically, the situation is even
worse. The DNA molecules coil the wrong way, and the mirror-"person"
could no more mate with a real person than could a nosrep!

But wait-you can get your heart to stay on the proper side if, instead, you flip
yourself head over heels, as if swinging over a waist-high horizontal bar in
front of you. Now your heart is on the same side as the mirror-person's
heart-but your feet and head are in the wrong places, and your stomach, although
at approximately the right height, is upsidedown. So it seems a mirror can be
perceived as reversing up and down, provided you're willing to map yourself onto
a creature whose feet are above its head. It all depends on the ways that you
are willing to slip yourself onto another entity. You have a choice of twirling
around a horizontal or a vertical bar, and getting the heart right but not the
head and feet, or getting the head and feet right but not the heart. It's simply
that, because of the external vertical symmetry of the human body, the vertical
self-twirling yields a more plausible-seeming you-to-image mapping. But mirrors
intrinsically don't care which way you interpret what they do. And in fact, all
they really reverse is back and front!

The well known psychologist and vision scientist
Richard Gregory (who
discovered and analysed many fascinating visual illusions, and founded the
Bristol
Exploratory,
among other things),
wrote a short review of
The Mind's I
in New Scientist December 1981,
available online
here.

In column 4 he criticises Hofstadter for not noticing that the left/right
rotation of text in a mirror occurs because the text has been rotated about its
vertical axis to face the mirror "giving left right reversal" whereas
"twiddling" text around its horizontal axis gives up-down and not right-left
reversal, adding "This has nothing whatever to do with mental mapping as
Hofstadter claims it has on page 404". It is not clear whether he interpreted
Hofstadter's analysis correctly.

A myth smashed on the floor

However, there is a very simple and obvious fact that appears not to have been
noticed by hundreds (thousands?) of people who worry about, or publish papers
about, the mirror puzzle. If, instead of thinking about real or imagined
rotations of the viewer, or real or imagined head-stands by the viewer, you just
put a mirror on the floor and stand next to it and look down, you'll find
yourself flipped upside down in the mirror. Moreover the upside down person,
will have front and back facing the same way as your front and back, unlike the
person reflected in the original vertical mirror.

Mirror, mirror on the floor,
Not flipped front-back any more??

The same happens if you attach a mirror to a ceiling, so that it is horizontal
and facing down. There is a restaurant in Vienna (or was when A.S. last visited
over 20 years ago) where the walls and some of the ceilings are covered in
mirrors, including the ceiling of the men's toilet above the urinals, producing
a somewhat disconcerting anti-gravity effect for anyone looking up!

So, to get the vertical flip, instead of rotating yourself, or imagining
rotating yourself, just rotate the mirror through 90 degrees until it is
horizontal, facing down from above, or facing up from below the viewer.

Note that if a mirror is not parallel to or perpendicular to a person's long
axis, then the reflection of the person will not be parallel to the person.

However, in all cases Hofstadter's point about asymmetric body parts not being
congruent with their mirror images is correct. E.g. in the reflection, your left
hand is congruent with your right hand outside the reflection. That is
independent of where the mirror is and where you are.

Ambulance Example

In the UK, and probably many other countries, an ambulance typically has the
word "Ambulance", or local equivalent, displayed
on the front in reverse, so that
car drivers will be able to read it correctly in their rear-view mirrors.

Here's the word "Ambulance" with no mirror to mess it around.

Notice that a vertical mirror will reverse the word "Ambulance" from left to
right even if the mirror is not facing the word, as sketched below (with
apologies for poor perspective drawing -- perhaps fixed later):

However a horizontal mirror will not reverse the text left to right, but top to
bottom, as depicted here:

Two Mirrors

The comparison is perhaps clearest with both mirrors present simultaneously.

Apologies for inaccurate geometrical relationships --
the reflections in the vertical mirror should be made to
tilt away to right, or the mirror should be made to appear
perpendicular to the wall, with the view-point changed.

And just to prove that it works with a real vertical mirror to one side:

Or with two mirrors: one on the floor, and one on the side:

Corner Example
(Added 19 Apr 2015)

The top-left square below is a photograph of a 3-D corner of a room, showing
portions of the floor and two walls, with three different objects indicating
three different (orthogonal) directions from the corner. The remaining squares
have three more versions of the same picture, with left and right reflected, up
and down reflected, and both left/right and up/down reflected, as indicated.

Each of the four pictures could be an original picture of a 3-D configuration.
Without the labels provided you could not tell which of the four is the
original, since the objects shown could be stuck to the surfaces, and therefore
information about gravity cannot be used to distinguish the pictures that have
been reflected up/down.

NOTE: all the reflections are reversible. E.g. the top right scene if
reflected horizontally becomes the top left scene. However, two scenes depicted
in the same row or the same column are, like a left and right hand, incapable of
being superimposed (even after rotations). They differ by one reflection. The
scenes that differ by two reflections, e.g. top left and bottom right, or top
right and bottom left, can be superimposed, like two left hands, or two right
hands.

Youtube presentation on the mirror puzzle

There is a delightful and very popular Youtube video (Physics Girl, presented
by Meg Chetwood), demonstrating what seems to be the theory proposed by Richard
Gregory, and emphasising the fact that humans have left-right symmetry but not
top bottom symmetry.
https://www.youtube.com/watch?v=vBpxhfBlVLU

The demonstration is excellent except for the fact that left-right symmetry has
nothing to do with what's going on, and neither does it matter how the thing to
be reflected in the mirror is presented (right way up, upside down, etc.)

The actual transformation produced depends only on the orientation of the
mirror, though how we think about it depends on how we describe ourselves, e.g.
as having a top and a bottom, a left side and a right side, a front and a back.

If the mirror is tilted at 45 degrees to a person's vertical axis it will
reflect a vertical person as a horizontal one, neither parallel to nor
perpendicular to the mirror.

What a mirror really does

Perhaps the most general way to think about what the mirror does is this: all
the surfaces visible from the mirror's reflective surface form a large hollow
container with a very complex shape. In producing the reflection, the mirror
turns that container inside out, something like a sock being pulled inside out,
except that a real sock may have different colours and textures on the inside
and the outside, whereas the imaginary sock turned inside out by the mirror has
exactly the same visible features on the inside as on the outside, which becomes
the inside after the reversal! This explains why the reflection of a left hand
is incongruent with the left hand, but congruent with the right hand. A
left-hand glove turned inside out fits on a right hand, and vice versa.
(This interpretation is due to Alastair Wilson.)

A familiar fact about psyche-d: a left hand glove will not fit on a right hand,
and vice versa, as Kant noted. Wittgenstein wrote in Tractatus 6.36111

A right-hand glove could be put on the left hand, if it
could be turned round in four-dimensional space.

But he does not seem to have noticed that even in 3-D space psyche-d can be
turned inside out, in which case a left hand glove will become able to fit on a
right hand, and vice versa.

It's possible that one of the commentators made these points in the many
comments on the video, but we have not searched through all of them.